Embedding problems in graphs and hypergraphs

Abstract

The first part of this thesis concerns perfect matchings and their generalisations. We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph, thereby answering a question of Hàn, Person and Schacht. We say that a graph \(G\) has a perfect \(H\)-packing (also called an \(H\) - factor) if there exists a set of disjoint copies of \(H\) in \(G\) which together cover all the vertices of \(G\). Given a graph \(H\), we determine, asymptotically, the Ore-type degree condition which ensures that a graph \(G\) has a perfect \(H\)-packing. The second part of the thesis concerns Hamilton cycles in directed graphs. We give a condition on the degree sequences of a digraph \(G\) that ensures \(G\) is Hamiltonian. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chvatal's theorem. We also show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \(\eta\) >0 every regular tournament \(G\) of sufficiently large order n contains at least (1/2- \(\eta\))n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968.

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